Review Reports

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The article contains interesting physicochemical research focused on determining the vapor pressure-mole fraction relationship of toluene and the predicted distillation curve for mixtures of organic compounds with different molecular structures. The presented research topic is interesting both scientifically and technologically and could provide a basis for further research on improving mixtures for the vapor pressure of aviation fuel. Below are a few minor comments and suggestions worth considering:

1. The abstract should be improved; it is too long. Please focus on describing the research objective, the methodology used, and the results obtained;
2. In the introduction, please describe the software currently used to determine vapor pressure – this mainly involves writing the software's proper names;
3. In the second chapter, it is worth adding a table with a summary of vapor pressure models – indicating the level of change – this will significantly improve the research topic and enhance readability;
4. In the subsection 3.1, please state whether the laboratory setup shown in Figure 1 is an original solution or a commercial one;
5. In the subsection 3.3, please state whether the temperature difference between the liquid in the boiler and the theoretical bubble temperature in the liquid mixture is within the measurement tolerance;
6. In the subsection 4.1, please add several statistical values, such as the standard deviation and confidence interval;
7. In the subsection 4.4 for Figure 11, please add an explanation for the horizontal lines – are these references to corrected measurements? Furthermore, please add two or three references that examined vapor pressure for aviation fuel distillation – to provide a comparison with previous studies;
8. The conclusions cover the scope of the analyses and are sufficient.

Author Response

Response to reviewers comments

Review 1

The article contains interesting physicochemical research focused on determining the vapor pressure-mole fraction relationship of toluene and the predicted distillation curve for mixtures of organic compounds with different molecular structures. The presented research topic is interesting both scientifically and technologically and could provide a basis for further research on improving mixtures for the vapor pressure of aviation fuel. Below are a few minor comments and suggestions worth considering:

 

1. The abstract should be improved; it is too long. Please focus on describing the research objective, the methodology used, and the results obtained;

Response:  Thank you for this input.  The word count has been reduced by 27% and the character count reduced by 35%.  The revised abstract more concisely highlights research objective and methodology, while continuing to direct the reader to the Results section to assess model validation.

 

2. In the introduction, please describe the software currently used to determine vapor pressure – this mainly involves writing the software's proper names;

Response:  Thank you for this input. The end of the 7^th paragraph has been revised to clarify that our implementation of UNIFAC was built into Excel, and it follows that our much simpler model is also amenable to any spreadsheet or home-grown application. The revised wording is presented here.

This model is called the UNIFAC model and is included in publicly available software such as Aspen Plus, DWSIM, and ChemCAD. It is simple enough to be implemented within a spreadsheet such as Excel™, which is what we did to enable comparisons with the new model presented in this work.  This spreadsheet is included in the supplemental material, entitled, “UNIFAC Vapor Pressures”. Yet more fundamental models based on intermolecular potentials (RGEMC) [13] or simulated osmotic pressure (OMD) [14] are available in molecular dynamics software such as LAMMPS, while REFPROP[15] employs mixing rules to develop an equation of state for the mixture[16] from which many properties can be calculated, including vapor pressure.

 

 

3. In the second chapter, it is worth adding a table with a summary of vapor pressure models – indicating the level of change – this will significantly improve the research topic and enhance readability;

Response:  A subsection called ‘Comparative summary of pressure models’ has been added to section 2. This subsection includes two paragraphs and a summary table. 

 

4. In the subsection 3.1, please state whether the laboratory setup shown in Figure 1 is an original solution or a commercial one;

Response:  The figure caption has been revised to clarify that this is a commercially available apparatus. The revised wording is presented here.

Figure 1. Instrument cross section.  Image taken from the Eralytics User Manual for the Eravap device which is commercially available.

 

5. In the subsection 3.3, please state whether the temperature difference between the liquid in the boiler and the theoretical bubble temperature in the liquid mixture is within the measurement tolerance;

Response:  Thank you for this input.  We agree that TC reproducibility should be noted and that none of the models predict the refluxing temperature within the uncertainty of the measurements.  However, we think this also fits in the second paragraph of subsection 4.2 (Modeled vapor pressure validation).  The revisions include a tweak to Figure 7 and the texts that are presented here.

Section 3.3)  The glassware shown in Figure 2 was used to measure the liquid temperature of a boiling mixture.  Type K thermocouples (±2.2 °C) were used to measure temperature. …

Section 4.2)  … As is quite evident from Figure 7, model 1D predicts a normal boiling point and refluxing temperature in far better agreement with the measured values than Raoult’s Law or the UNIFAC model. While none of the models predict the measured result to within the ±2.2 °C reproducibility of the TC, the vapor pressure predicted by model 1D at the lower edge of the measured temperature uncertainty band is just 11% higher than lab pressure; much closer than the other models.

Figure 7 caption:   Measured and predicted liquid temperature of a refluxing mixture of cyclohexane and o-xylene in tetradecane. The dashed blue lines correspond with the temperature measurement uncertainty ±2.2 °C. The dashed yellow lines correspond to the predicted temperature at a vapor pressure ±5% of lab pressure.

6. In the subsection 4.1, please add several statistical values, such as the standard deviation and confidence interval;

Response:  This is a good point. The text of subsection 4.1 already had rms and MAE for model 1D, but now we have added MARE for model 1D, Raoult’s Law, and UNIFAC.

 

7. In the subsection 4.4 for Figure 11, please add an explanation for the horizontal lines – are these references to corrected measurements? Furthermore, please add two or three references that examined vapor pressure for aviation fuel distillation – to provide a comparison with previous studies;

Response:  The captions to Figures 9 and 11 have now been revised to include an explanation for the horizonal uncertainty bars.  The revised text is provided below. In response to your 2^nd point, 3 additional references have been cited and noted below. 

Figure 9. Measured and predicted distillation curve of a mixture of cyclohexane and o-xylene in tetradecane. Filled symbols correspond to measurements. Open symbols correspond to corrected measurements. The uncertainty bars correspond to the range of our estimated dynamic hold up volume. Curves correspond to model predictions.

Figure 11. Measured and predicted distillation curve of a mixture of isooctane and o-xylene in tetradecane. Filled symbols correspond to measurements. Open symbols correspond to corrected measurements. The uncertainty bars correspond to the range of our estimated dynamic hold up volume. Curves and x’s correspond to model predictions.

[24]      Opacich KC, Peiffer E, Heyne JS. Analyzing the Relative Impact of Spray and Volatile Fuel Properties on Gas Turbine Combustor Ignition in Multiple Rig Geometries 2019:1–10. https://doi.org/10.2514/6.2019-1434.

[25]      Rauch B. Systematic accuracy assessment for alternative aviation fuel evaporation models. Universität Stuttgart, 2017. https://elib.dlr.de/123687/1/2018_BRauch_Dissertation_Systematic_Accuracy_Assessment_Alternative_Fuel_Evaporation_Models.pdf

[34]      Yang Z, Boehm RC, Bell DC, Heyne JS. Towards sustainable aviation fuel distillation optimization. Fuel 2023;353:129136. https://doi.org/doi.org/10.1016/j.fuel.2023.129136.

 

8. The conclusions cover the scope of the analyses and are sufficient

 

Review 2

This study aims to develop a simplified model for accurately predicting the vapor pressure of multicomponent hydrocarbon mixtures typical of jet fuels. The authors note that classical Raoult's law is inapplicable at low mole fractions of the components, while more complex models, such as UNIFAC, are computationally intensive and have not been adequately validated for concentrations below 10 mol%. A modification of the Clausius-Clapeyron equation is proposed as a solution, replacing the reference temperature and enthalpy of vaporization with simple linear functions of the component mole fraction. The model, designated Model 1D, contains two empirical parameters that were tuned using experimental vapor pressure data for binary mixtures of n-pentane/n-dodecane and toluene/n-dodecane at 70 and 100°C. The new model was shown to predict vapor pressure significantly more accurately than Raoult's law, demonstrating accuracy comparable to or superior to the UNIFAC model while possessing a significantly simpler mathematical form. Distillation experiments confirmed the superior predictive power of the 1D model compared to Raoult's law, but also revealed that the error in modeling the distillation process itself (e.g., determining the number of theoretical plates) can be comparable to or greater than the error in the vapor pressure model. The main advantage of the proposed model is its computational efficiency, making it promising for use in multiscale modeling, including evaporation calculations in combustion chambers, which require multiple vapor pressure calculations for thousands of components.

However, it would be necessary to clarify a number of comments that are available to the article:

 

1. The model is tuned to only 16 experimental points for two binary systems (n-pentane/n-dodecane and toluene/n-dodecane) at two temperatures (70 and 100°C). This raises questions about its applicability to a wider range of hydrocarbons (e.g., cycloparaffins, olefins) and over a wider range of temperatures and pressures.

Response:  This is a reasonable concern. Only through additional validation will confidence in and thorough understanding of its scope of applicability to be gained. In this work we do include other molecules in the validation suite; including cyclohexane, methylcyclopentane, methylcyclohexane, isooctane, n-octane, n-nonane, n-tetradecane, and o-xylene. We cover a pressure range from 1.8 to 101 kPa, and have now added additional text in subsection 3.2 (which has been renamed as ‘Materials and temperatures’) to clearly state that we do not believe the model should be applied to high pressures, where real-gas effects are important. This revised text reads as follows:

… Their study involved five sets of binary mixtures involving n-nonane, n-octane, methylcyclohexane and methylcyclopentane and three fixed temperatures (120, 160, 200 °C). We selected the points at 120 °C to minimize the confounding effects of real gas behavior, as model 1D does not address heterogeneous intermolecular attractions/repulsions in the gas phase.”

Additionally, the newly added subsection, 2.4 Comparative summary of pressure models, discusses this point and a sentence has been added to the 1^st paragraph of subsection 4.1 to address your comment about cycloalkanes and alkenes. The revised text is presented here.

 … the entropy of vaporization, which effectively relates the normal boiling point temperature and the heat of vaporization, is somewhat different for each; approximately 86.5 and 72.5 J/mol/K for alkanes and aromatics respectively. Our intent here is to model each in the same way and to assess the error introduced by neglecting this difference.  For reference, the approximate entropy of vaporization of cycloalkanes and alkenes are 85.0 and 82.5 J/mol/K respectively, within the bounds of the representatives chosen for this work.

 

2. The statement that the 1D model shows a mean absolute relative error (MARE) of 5.1% on the data of Hung et al. requires context. It is stated that for UNIFAC-DMD, the MARE is 1.5%. Therefore, the 1D model is significantly inferior to the specialized UNIFAC-DMD model on these data. This is an important limitation that should be emphasized, rather than simply stating comparability with COSMO-SAC.

Response:  This is a good point.  We want to emphasize the algebraic simplicity of model 1D, and not present it as being generally more accurate than UNIFAC. Our goal from the on-set was to develop a hand-calc-level model (suitable for spreadsheets, homegrown python codes with this calc inserted into the innermost loop, or user-defined functions in CFD codes) that is clearly more accurate than Raoult’s law and competitive with more complicated models when applied to mixtures like sustainable aviation fuel or its unrefined precursors at conditions consistent with refining or spays in jet-engine combustors (or fuel systems) when operating at ignition or flight idle conditions. Thus, a paragraph has been added to subsection 4.3, Literature data, to highlight this point.  The revised text is presented below;

… In contrast, Raoult’s Law over-predicts the vapor pressure of all 20 mixtures; two of them by more than 40% and has a MARE of 13.7%.

It is worth noting here that the performance of model 1D on this dataset is consistent with the authors original intent which was to develop an algebraically simple model (suitable for hand calcs, spreadsheets, or codes/applications that require vapor pressure or composition many thousands of times per run) that provides much improved accuracy relative to Raoult’s Law while remaining comparatively accurate to the more complicated and mature models used in commercially available applications that serve process engineering and industrial simulations. Model 1D is not intended to replace models such as UNIFAC but rather to reach applications (through its simplicity) that are not currently pragmatic for UNIFAC. Specifically, this model does not address heterogeneous, real gas effects and should not be applied to conditions where such terms are expected to be important. It can be applied to our research goals which are to support to support conceptual-level distillation cut-point optimization such as exemplified by Yang et al. [2023], bubble point prediction in the fuel system at chop from cruise to flight idle, preferential evaporation effects on lean blow at this same (jet engine) operating condition, and evaporation rate predictions in (future) simulations of ignition in jet engines.  Of these, the only one that is pragmatic for applications like ASPEN is the check on bubble point in the fuel system at the min-thrust / max-altitude (combustor) design point, but jet engine combustor designers do not generally have access to any such commercial application.

 

3. In Section 4.1, it is stated that for a mixture with 9.0 mol. For n-pentane at 100°C, the UNIFAC error reaches 31%. However, similar quantitative comparisons of errors for the 1D model on the same data are not presented in Sections 4.2 or 4.3. A direct quantitative evaluation (e.g., tabulated with RMSE, MARE) is required for all models on all validation datasets.

Response:  The following sentences have been added/amended to the first paragraphs of subsections 4.2 and 4.3 respectively.  Additionally, edits have been made around Figure 7 (in subsection 4.2) and Figure 8 (in subsection 4.3) as noted elsewhere in this document.

4.2) …  In numerical summary, the MARE of model 1D, Raoult’s Law, UNIFAC and REFPROP are 44.6%, 24.3%, 33.6%, and 5.5% respectively. Only model 1D has a lower MRE magnitude (5.1%) than MARE.

4.3) … In contrast, Raoult’s Law over-predicts the vapor pressure of all 20 mixtures; two of them by more than 40% and has a MARE of 13.7%.

 

 

4. Figures 9–12 demonstrate that the distillation process modeling error (the number of theoretical plates) dominates the vapor pressure model error. This calls into question the feasibility of validating the vapor pressure model solely from distillation curves. The conclusion that the 1D model "exceeds" Raoult's law is based on visual similarity, while the RMS errors remain high (9.2–12.3°C).

Response:  We agree that the distillation curves offer little validation of the vapor pressure model. The distillate compositions are much more revealing.  Whether the composition comparison is done at a best-match integer-plate distillation assumption or an optimized non-integer-plate distillation assumption, the distillate composition predicted by model 1D is much closer to the measured distillate composition than what results from Raoult’s Law. We do not understand what should be changed, if anything, in the manuscript to address this comment.

 

5. The error bar chart is informative, but it would be useful to add a target region (e.g., ±5%) to the plot to visually assess the performance of the models (Figure 8).

Response:  It is not entirely clear to us that you are referring to Figure 8 (as stated) or the bar chart, Figure 7.  A shaded region corresponding to ±5% relative error has been added to Figure 8, with the caption amended accordingly – as presented here.

Figure 7. Measured and predicted liquid temperature of a refluxing mixture of cyclohexane and o-xylene in tetradecane. The dashed blue lines correspond with the temperature measurement uncertainty ±2.2 °C. The dashed yellow lines correspond to the predicted temperature at a vapor pressure ±5% of lab pressure.

 

6. In the distillation experiments, the distillate collection rate (1.5-3.0 ml/min) was lower than that required by ASTM D86 (4-5 ml/min). This deviation from the standard could have affected the degree of fractionation and, consequently, the results, which requires a more detailed discussion.

Response:  This is a good point. A sentence has been added to subsection 3.4 to make this point more apparent to all readers.

Distillations were carried out with the apparatus shown in Figure 3, achieving a collection rate of 1.5-3.0 mL/min from 5% distilled to 50% distilled, which is lower than the requirement stated in ASTM D86 (4-5 mL/min).  As a consequence, the degree of separation attained in this set-up is likely somewhat higher than is typical of apparatuses used in conjunction with ASTM D86.

 

7. The model parameters (s for B and C) are presented as empirical. Their physical meaning and how their values (e.g., the decrease in effective activation energy with decreasing mole fraction) relate to solution theory are not discussed in sufficient depth.

 

Response:  The paragraph presented below has been inserted to subsection 2.1 immediately after the 4^th paragraph, where we had compared heterogeneous neighbors with vacancies, where the latter gives rise to the well-known temperature impact on the desorption rate activation energy.  Additionally, a sentence has been inserted in the last paragraph of this subsection.

New paragraph) For a more fundamental description of the relationship between the molecular arrangements in the condensed phase and mobility of molecules or vacancies within the condensed phase the reader is referred to Boehm et al. [23]. Briefly, in the condensed phase (the liquid), the desorbing molecule from a mixture sometimes originates (departs) from a shallower (potential energy surface) well than it would if it were not mixed with heterogeneous molecules, but essentially never originates from a deeper well. This is because optimal molecular packing in the condensed phase occurs when all the molecules are the same size and solutes are essentially forced into a packing arrangement that is determined by the solvent. The corresponding transition state, however is less impacted by the mixture-induced geometric distortions of the condensed-phase lattice. This results in a lower effective activation energy. Quantifying this energy difference from first principles would be very difficult. Instead, we simply acknowledge that it should be lower and dependent on mole fraction and use measured vapor pressures to gage its magnitude.

Added sentence) … but also because the reference temperature  (, the temperature at which   atm) varies as the effective surface area of the i^th component, its concentration, varies. The desorption rate scales with mole fraction (surface coverage) while the adsorption rate does not.

 

 

 

 

Editor

I am writing to discuss several issues raised in the manuscript you submitted. Please note the following two points when revising your manuscript:

1. Sustainability Connection

After the Editorial Board's review, we found that the connection between your manuscript and sustainability is relatively weak. During the revision process of your manuscript, please strengthen the connection with sustainability, especially in the abstract, keywords, and the

introductory part of the article. This is just a gentle reminder to consider adding more terms related to "sustain" (e.g., "sustainable" and "sustainability") where appropriate.

Response:  A keyword, “NetZero Aviation”, a paragraph (shown below) has been added to the introduction, and we had tried to emphasize aspects of sustainability throughout the revisions.

SAF and synthetic blending components (SBCs) are considered mid-term solutions to help decarbonize the aviation sector by 2050.[2] The term SAF is primarily used in policy and regulatory contexts, whereas SBC is the corresponding technical term used in fuel qualification standards. Currently, SBCs are limited up to 50% by volume when blended with conventional jet fuel because some pathways (e.g., HEFA-SPK) lack certain hydrocarbon classes such as aromatics, which are necessary for elastomer swelling and ensuring proper fuel system performance. To overcome this limitation and enable 100% synthetic ‘drop-in’ jet fuels, the aviation community is exploring blending of multiple ASTM D7566 annexes. However, original equipment manufacturers (OEMs), prioritizing safety and reliability, have emphasized the need to evaluate ASTM D4054 Tier 2, or fit-for-purpose (FFP), properties to ensure that fully synthetic fuels meet operational requirements. Measuring all FFP properties experimentally is costly and often beyond the capabilities of many refineries. As a result, the ability to accurately predict FFP properties, particularly volatility and vapor pressure has become a high priority within the fuel community, as it would accelerate certification, enable higher SAF blend limits, and ultimately support aviation decarbonization efforts.

 

2. Invalid URL

We noted that the links for References 3 and 12 are currently not accessible. We kindly ask you to verify these URLs and provide working ones. Please ensure that all webpages cited in the manuscript are fully accessible.

Response:  Thank you for finding this issue.  The URL’s have been corrected.

 

 

Reviewer 2 Report

Comments and Suggestions for Authors

This study aims to develop a simplified model for accurately predicting the vapor pressure of multicomponent hydrocarbon mixtures typical of jet fuels. The authors note that classical Raoult's law is inapplicable at low mole fractions of the components, while more complex models, such as UNIFAC, are computationally intensive and have not been adequately validated for concentrations below 10 mol%. A modification of the Clausius-Clapeyron equation is proposed as a solution, replacing the reference temperature and enthalpy of vaporization with simple linear functions of the component mole fraction. The model, designated Model 1D, contains two empirical parameters that were tuned using experimental vapor pressure data for binary mixtures of n-pentane/n-dodecane and toluene/n-dodecane at 70 and 100°C. The new model was shown to predict vapor pressure significantly more accurately than Raoult's law, demonstrating accuracy comparable to or superior to the UNIFAC model while possessing a significantly simpler mathematical form. Distillation experiments confirmed the superior predictive power of the 1D model compared to Raoult's law, but also revealed that the error in modeling the distillation process itself (e.g., determining the number of theoretical plates) can be comparable to or greater than the error in the vapor pressure model. The main advantage of the proposed model is its computational efficiency, making it promising for use in multiscale modeling, including evaporation calculations in combustion chambers, which require multiple vapor pressure calculations for thousands of components.

 

However, it would be necessary to clarify a number of comments that are available to the article:

1. The model is tuned to only 16 experimental points for two binary systems (n-pentane/n-dodecane and toluene/n-dodecane) at two temperatures (70 and 100°C). This raises questions about its applicability to a wider range of hydrocarbons (e.g., cycloparaffins, olefins) and over a wider range of temperatures and pressures.
2. The statement that the 1D model shows a mean absolute relative error (MARE) of 5.1% on the data of Hung et al. requires context. It is stated that for UNIFAC-DMD, the MARE is 1.5%. Therefore, the 1D model is significantly inferior to the specialized UNIFAC-DMD model on these data. This is an important limitation that should be emphasized, rather than simply stating comparability with COSMO-SAC.
3. In Section 4.1, it is stated that for a mixture with 9.0 mol. For n-pentane at 100°C, the UNIFAC error reaches 31%. However, similar quantitative comparisons of errors for the 1D model on the same data are not presented in Sections 4.2 or 4.3. A direct quantitative evaluation (e.g., tabulated with RMSE, MARE) is required for all models on all validation datasets.
4. Figures 9–12 demonstrate that the distillation process modeling error (the number of theoretical plates) dominates the vapor pressure model error. This calls into question the feasibility of validating the vapor pressure model solely from distillation curves. The conclusion that the 1D model "exceeds" Raoult's law is based on visual similarity, while the RMS errors remain high (9.2–12.3°C).
5. The error bar chart is informative, but it would be useful to add a target region (e.g., ±5%) to the plot to visually assess the performance of the models (Figure 8).
6. In the distillation experiments, the distillate collection rate (1.5-3.0 ml/min) was lower than that required by ASTM D86 (4-5 ml/min). This deviation from the standard could have affected the degree of fractionation and, consequently, the results, which requires a more detailed discussion.
7. The model parameters (s for B and C) are presented as empirical. Their physical meaning and how their values ​​(e.g., the decrease in effective activation energy with decreasing mole fraction) relate to solution theory are not discussed in sufficient depth.

Author Response

Response to reviewers comments

Review 1

The article contains interesting physicochemical research focused on determining the vapor pressure-mole fraction relationship of toluene and the predicted distillation curve for mixtures of organic compounds with different molecular structures. The presented research topic is interesting both scientifically and technologically and could provide a basis for further research on improving mixtures for the vapor pressure of aviation fuel. Below are a few minor comments and suggestions worth considering:

 

1. The abstract should be improved; it is too long. Please focus on describing the research objective, the methodology used, and the results obtained;

Response:  Thank you for this input.  The word count has been reduced by 27% and the character count reduced by 35%.  The revised abstract more concisely highlights research objective and methodology, while continuing to direct the reader to the Results section to assess model validation.

 

2. In the introduction, please describe the software currently used to determine vapor pressure – this mainly involves writing the software's proper names;

Response:  Thank you for this input. The end of the 7^th paragraph has been revised to clarify that our implementation of UNIFAC was built into Excel, and it follows that our much simpler model is also amenable to any spreadsheet or home-grown application. The revised wording is presented here.

This model is called the UNIFAC model and is included in publicly available software such as Aspen Plus, DWSIM, and ChemCAD. It is simple enough to be implemented within a spreadsheet such as Excel™, which is what we did to enable comparisons with the new model presented in this work.  This spreadsheet is included in the supplemental material, entitled, “UNIFAC Vapor Pressures”. Yet more fundamental models based on intermolecular potentials (RGEMC) [13] or simulated osmotic pressure (OMD) [14] are available in molecular dynamics software such as LAMMPS, while REFPROP[15] employs mixing rules to develop an equation of state for the mixture[16] from which many properties can be calculated, including vapor pressure.

 

 

3. In the second chapter, it is worth adding a table with a summary of vapor pressure models – indicating the level of change – this will significantly improve the research topic and enhance readability;

Response:  A subsection called ‘Comparative summary of pressure models’ has been added to section 2. This subsection includes two paragraphs and a summary table. 

 

4. In the subsection 3.1, please state whether the laboratory setup shown in Figure 1 is an original solution or a commercial one;

Response:  The figure caption has been revised to clarify that this is a commercially available apparatus. The revised wording is presented here.

Figure 1. Instrument cross section.  Image taken from the Eralytics User Manual for the Eravap device which is commercially available.

 

5. In the subsection 3.3, please state whether the temperature difference between the liquid in the boiler and the theoretical bubble temperature in the liquid mixture is within the measurement tolerance;

Response:  Thank you for this input.  We agree that TC reproducibility should be noted and that none of the models predict the refluxing temperature within the uncertainty of the measurements.  However, we think this also fits in the second paragraph of subsection 4.2 (Modeled vapor pressure validation).  The revisions include a tweak to Figure 7 and the texts that are presented here.

Section 3.3)  The glassware shown in Figure 2 was used to measure the liquid temperature of a boiling mixture.  Type K thermocouples (±2.2 °C) were used to measure temperature. …

Section 4.2)  … As is quite evident from Figure 7, model 1D predicts a normal boiling point and refluxing temperature in far better agreement with the measured values than Raoult’s Law or the UNIFAC model. While none of the models predict the measured result to within the ±2.2 °C reproducibility of the TC, the vapor pressure predicted by model 1D at the lower edge of the measured temperature uncertainty band is just 11% higher than lab pressure; much closer than the other models.

Figure 7 caption:   Measured and predicted liquid temperature of a refluxing mixture of cyclohexane and o-xylene in tetradecane. The dashed blue lines correspond with the temperature measurement uncertainty ±2.2 °C. The dashed yellow lines correspond to the predicted temperature at a vapor pressure ±5% of lab pressure.

6. In the subsection 4.1, please add several statistical values, such as the standard deviation and confidence interval;

Response:  This is a good point. The text of subsection 4.1 already had rms and MAE for model 1D, but now we have added MARE for model 1D, Raoult’s Law, and UNIFAC.

 

7. In the subsection 4.4 for Figure 11, please add an explanation for the horizontal lines – are these references to corrected measurements? Furthermore, please add two or three references that examined vapor pressure for aviation fuel distillation – to provide a comparison with previous studies;

Response:  The captions to Figures 9 and 11 have now been revised to include an explanation for the horizonal uncertainty bars.  The revised text is provided below. In response to your 2^nd point, 3 additional references have been cited and noted below. 

Figure 9. Measured and predicted distillation curve of a mixture of cyclohexane and o-xylene in tetradecane. Filled symbols correspond to measurements. Open symbols correspond to corrected measurements. The uncertainty bars correspond to the range of our estimated dynamic hold up volume. Curves correspond to model predictions.

Figure 11. Measured and predicted distillation curve of a mixture of isooctane and o-xylene in tetradecane. Filled symbols correspond to measurements. Open symbols correspond to corrected measurements. The uncertainty bars correspond to the range of our estimated dynamic hold up volume. Curves and x’s correspond to model predictions.

[24]      Opacich KC, Peiffer E, Heyne JS. Analyzing the Relative Impact of Spray and Volatile Fuel Properties on Gas Turbine Combustor Ignition in Multiple Rig Geometries 2019:1–10. https://doi.org/10.2514/6.2019-1434.

[25]      Rauch B. Systematic accuracy assessment for alternative aviation fuel evaporation models. Universität Stuttgart, 2017. https://elib.dlr.de/123687/1/2018_BRauch_Dissertation_Systematic_Accuracy_Assessment_Alternative_Fuel_Evaporation_Models.pdf

[34]      Yang Z, Boehm RC, Bell DC, Heyne JS. Towards sustainable aviation fuel distillation optimization. Fuel 2023;353:129136. https://doi.org/doi.org/10.1016/j.fuel.2023.129136.

 

8. The conclusions cover the scope of the analyses and are sufficient

 

Review 2

This study aims to develop a simplified model for accurately predicting the vapor pressure of multicomponent hydrocarbon mixtures typical of jet fuels. The authors note that classical Raoult's law is inapplicable at low mole fractions of the components, while more complex models, such as UNIFAC, are computationally intensive and have not been adequately validated for concentrations below 10 mol%. A modification of the Clausius-Clapeyron equation is proposed as a solution, replacing the reference temperature and enthalpy of vaporization with simple linear functions of the component mole fraction. The model, designated Model 1D, contains two empirical parameters that were tuned using experimental vapor pressure data for binary mixtures of n-pentane/n-dodecane and toluene/n-dodecane at 70 and 100°C. The new model was shown to predict vapor pressure significantly more accurately than Raoult's law, demonstrating accuracy comparable to or superior to the UNIFAC model while possessing a significantly simpler mathematical form. Distillation experiments confirmed the superior predictive power of the 1D model compared to Raoult's law, but also revealed that the error in modeling the distillation process itself (e.g., determining the number of theoretical plates) can be comparable to or greater than the error in the vapor pressure model. The main advantage of the proposed model is its computational efficiency, making it promising for use in multiscale modeling, including evaporation calculations in combustion chambers, which require multiple vapor pressure calculations for thousands of components.

However, it would be necessary to clarify a number of comments that are available to the article:

 

1. The model is tuned to only 16 experimental points for two binary systems (n-pentane/n-dodecane and toluene/n-dodecane) at two temperatures (70 and 100°C). This raises questions about its applicability to a wider range of hydrocarbons (e.g., cycloparaffins, olefins) and over a wider range of temperatures and pressures.

Response:  This is a reasonable concern. Only through additional validation will confidence in and thorough understanding of its scope of applicability to be gained. In this work we do include other molecules in the validation suite; including cyclohexane, methylcyclopentane, methylcyclohexane, isooctane, n-octane, n-nonane, n-tetradecane, and o-xylene. We cover a pressure range from 1.8 to 101 kPa, and have now added additional text in subsection 3.2 (which has been renamed as ‘Materials and temperatures’) to clearly state that we do not believe the model should be applied to high pressures, where real-gas effects are important. This revised text reads as follows:

… Their study involved five sets of binary mixtures involving n-nonane, n-octane, methylcyclohexane and methylcyclopentane and three fixed temperatures (120, 160, 200 °C). We selected the points at 120 °C to minimize the confounding effects of real gas behavior, as model 1D does not address heterogeneous intermolecular attractions/repulsions in the gas phase.”

Additionally, the newly added subsection, 2.4 Comparative summary of pressure models, discusses this point and a sentence has been added to the 1^st paragraph of subsection 4.1 to address your comment about cycloalkanes and alkenes. The revised text is presented here.

 … the entropy of vaporization, which effectively relates the normal boiling point temperature and the heat of vaporization, is somewhat different for each; approximately 86.5 and 72.5 J/mol/K for alkanes and aromatics respectively. Our intent here is to model each in the same way and to assess the error introduced by neglecting this difference.  For reference, the approximate entropy of vaporization of cycloalkanes and alkenes are 85.0 and 82.5 J/mol/K respectively, within the bounds of the representatives chosen for this work.

 

2. The statement that the 1D model shows a mean absolute relative error (MARE) of 5.1% on the data of Hung et al. requires context. It is stated that for UNIFAC-DMD, the MARE is 1.5%. Therefore, the 1D model is significantly inferior to the specialized UNIFAC-DMD model on these data. This is an important limitation that should be emphasized, rather than simply stating comparability with COSMO-SAC.

Response:  This is a good point.  We want to emphasize the algebraic simplicity of model 1D, and not present it as being generally more accurate than UNIFAC. Our goal from the on-set was to develop a hand-calc-level model (suitable for spreadsheets, homegrown python codes with this calc inserted into the innermost loop, or user-defined functions in CFD codes) that is clearly more accurate than Raoult’s law and competitive with more complicated models when applied to mixtures like sustainable aviation fuel or its unrefined precursors at conditions consistent with refining or spays in jet-engine combustors (or fuel systems) when operating at ignition or flight idle conditions. Thus, a paragraph has been added to subsection 4.3, Literature data, to highlight this point.  The revised text is presented below;

… In contrast, Raoult’s Law over-predicts the vapor pressure of all 20 mixtures; two of them by more than 40% and has a MARE of 13.7%.

It is worth noting here that the performance of model 1D on this dataset is consistent with the authors original intent which was to develop an algebraically simple model (suitable for hand calcs, spreadsheets, or codes/applications that require vapor pressure or composition many thousands of times per run) that provides much improved accuracy relative to Raoult’s Law while remaining comparatively accurate to the more complicated and mature models used in commercially available applications that serve process engineering and industrial simulations. Model 1D is not intended to replace models such as UNIFAC but rather to reach applications (through its simplicity) that are not currently pragmatic for UNIFAC. Specifically, this model does not address heterogeneous, real gas effects and should not be applied to conditions where such terms are expected to be important. It can be applied to our research goals which are to support to support conceptual-level distillation cut-point optimization such as exemplified by Yang et al. [2023], bubble point prediction in the fuel system at chop from cruise to flight idle, preferential evaporation effects on lean blow at this same (jet engine) operating condition, and evaporation rate predictions in (future) simulations of ignition in jet engines.  Of these, the only one that is pragmatic for applications like ASPEN is the check on bubble point in the fuel system at the min-thrust / max-altitude (combustor) design point, but jet engine combustor designers do not generally have access to any such commercial application.

 

3. In Section 4.1, it is stated that for a mixture with 9.0 mol. For n-pentane at 100°C, the UNIFAC error reaches 31%. However, similar quantitative comparisons of errors for the 1D model on the same data are not presented in Sections 4.2 or 4.3. A direct quantitative evaluation (e.g., tabulated with RMSE, MARE) is required for all models on all validation datasets.

Response:  The following sentences have been added/amended to the first paragraphs of subsections 4.2 and 4.3 respectively.  Additionally, edits have been made around Figure 7 (in subsection 4.2) and Figure 8 (in subsection 4.3) as noted elsewhere in this document.

4.2) …  In numerical summary, the MARE of model 1D, Raoult’s Law, UNIFAC and REFPROP are 44.6%, 24.3%, 33.6%, and 5.5% respectively. Only model 1D has a lower MRE magnitude (5.1%) than MARE.

4.3) … In contrast, Raoult’s Law over-predicts the vapor pressure of all 20 mixtures; two of them by more than 40% and has a MARE of 13.7%.

 

 

4. Figures 9–12 demonstrate that the distillation process modeling error (the number of theoretical plates) dominates the vapor pressure model error. This calls into question the feasibility of validating the vapor pressure model solely from distillation curves. The conclusion that the 1D model "exceeds" Raoult's law is based on visual similarity, while the RMS errors remain high (9.2–12.3°C).

Response:  We agree that the distillation curves offer little validation of the vapor pressure model. The distillate compositions are much more revealing.  Whether the composition comparison is done at a best-match integer-plate distillation assumption or an optimized non-integer-plate distillation assumption, the distillate composition predicted by model 1D is much closer to the measured distillate composition than what results from Raoult’s Law. We do not understand what should be changed, if anything, in the manuscript to address this comment.

 

5. The error bar chart is informative, but it would be useful to add a target region (e.g., ±5%) to the plot to visually assess the performance of the models (Figure 8).

Response:  It is not entirely clear to us that you are referring to Figure 8 (as stated) or the bar chart, Figure 7.  A shaded region corresponding to ±5% relative error has been added to Figure 8, with the caption amended accordingly – as presented here.

Figure 7. Measured and predicted liquid temperature of a refluxing mixture of cyclohexane and o-xylene in tetradecane. The dashed blue lines correspond with the temperature measurement uncertainty ±2.2 °C. The dashed yellow lines correspond to the predicted temperature at a vapor pressure ±5% of lab pressure.

 

6. In the distillation experiments, the distillate collection rate (1.5-3.0 ml/min) was lower than that required by ASTM D86 (4-5 ml/min). This deviation from the standard could have affected the degree of fractionation and, consequently, the results, which requires a more detailed discussion.

Response:  This is a good point. A sentence has been added to subsection 3.4 to make this point more apparent to all readers.

Distillations were carried out with the apparatus shown in Figure 3, achieving a collection rate of 1.5-3.0 mL/min from 5% distilled to 50% distilled, which is lower than the requirement stated in ASTM D86 (4-5 mL/min).  As a consequence, the degree of separation attained in this set-up is likely somewhat higher than is typical of apparatuses used in conjunction with ASTM D86.

 

7. The model parameters (s for B and C) are presented as empirical. Their physical meaning and how their values (e.g., the decrease in effective activation energy with decreasing mole fraction) relate to solution theory are not discussed in sufficient depth.

 

Response:  The paragraph presented below has been inserted to subsection 2.1 immediately after the 4^th paragraph, where we had compared heterogeneous neighbors with vacancies, where the latter gives rise to the well-known temperature impact on the desorption rate activation energy.  Additionally, a sentence has been inserted in the last paragraph of this subsection.

New paragraph) For a more fundamental description of the relationship between the molecular arrangements in the condensed phase and mobility of molecules or vacancies within the condensed phase the reader is referred to Boehm et al. [23]. Briefly, in the condensed phase (the liquid), the desorbing molecule from a mixture sometimes originates (departs) from a shallower (potential energy surface) well than it would if it were not mixed with heterogeneous molecules, but essentially never originates from a deeper well. This is because optimal molecular packing in the condensed phase occurs when all the molecules are the same size and solutes are essentially forced into a packing arrangement that is determined by the solvent. The corresponding transition state, however is less impacted by the mixture-induced geometric distortions of the condensed-phase lattice. This results in a lower effective activation energy. Quantifying this energy difference from first principles would be very difficult. Instead, we simply acknowledge that it should be lower and dependent on mole fraction and use measured vapor pressures to gage its magnitude.

Added sentence) … but also because the reference temperature  (, the temperature at which   atm) varies as the effective surface area of the i^th component, its concentration, varies. The desorption rate scales with mole fraction (surface coverage) while the adsorption rate does not.

 

 

 

 

Editor

I am writing to discuss several issues raised in the manuscript you submitted. Please note the following two points when revising your manuscript:

1. Sustainability Connection

After the Editorial Board's review, we found that the connection between your manuscript and sustainability is relatively weak. During the revision process of your manuscript, please strengthen the connection with sustainability, especially in the abstract, keywords, and the

introductory part of the article. This is just a gentle reminder to consider adding more terms related to "sustain" (e.g., "sustainable" and "sustainability") where appropriate.

Response:  A keyword, “NetZero Aviation”, a paragraph (shown below) has been added to the introduction, and we had tried to emphasize aspects of sustainability throughout the revisions.

SAF and synthetic blending components (SBCs) are considered mid-term solutions to help decarbonize the aviation sector by 2050.[2] The term SAF is primarily used in policy and regulatory contexts, whereas SBC is the corresponding technical term used in fuel qualification standards. Currently, SBCs are limited up to 50% by volume when blended with conventional jet fuel because some pathways (e.g., HEFA-SPK) lack certain hydrocarbon classes such as aromatics, which are necessary for elastomer swelling and ensuring proper fuel system performance. To overcome this limitation and enable 100% synthetic ‘drop-in’ jet fuels, the aviation community is exploring blending of multiple ASTM D7566 annexes. However, original equipment manufacturers (OEMs), prioritizing safety and reliability, have emphasized the need to evaluate ASTM D4054 Tier 2, or fit-for-purpose (FFP), properties to ensure that fully synthetic fuels meet operational requirements. Measuring all FFP properties experimentally is costly and often beyond the capabilities of many refineries. As a result, the ability to accurately predict FFP properties, particularly volatility and vapor pressure has become a high priority within the fuel community, as it would accelerate certification, enable higher SAF blend limits, and ultimately support aviation decarbonization efforts.

 

2. Invalid URL

We noted that the links for References 3 and 12 are currently not accessible. We kindly ask you to verify these URLs and provide working ones. Please ensure that all webpages cited in the manuscript are fully accessible.

Response:  Thank you for finding this issue.  The URL’s have been corrected.

 

 

Round 2

Reviewer 2 Report

Comments and Suggestions for Authors

This paper addresses a pressing issue: developing a simplified model for predicting vapor pressure and phase equilibria in complex multicomponent mixtures such as aviation fuels.

1. Model applicability. A well-founded concern was expressed that the model was tuned to a narrow data set (16 data points for two binary systems). In response, the authors significantly expanded the validation section to include data on cycloalkanes (cyclohexane, methylcyclopentane, and methylcyclohexane), olefins (via reference to the entropy of vaporization), and other hydrocarbons (isooctane, n-octane, and n-nonane). The model's limitation to low-pressure conditions, where real-gas effects are insignificant, was explicitly stated. These additions adequately address the criticism and clarify the model's applicability.
2. Comparison with UNIFAC-DMD. The criticism that the comparison with UNIFAC-DMD (MARE 1.5%) was not fully presented has been completely addressed. The authors added an important paragraph to Section 4.3, clearly articulating the philosophy of their work: the goal is not to replace complex industrial models, but to create a simple tool for applications where UNIFAC is impractical. This is a fair and reasonable position, placing the work in the correct context.
3. Quantitative comparison of errors. The authors satisfied the requirement for a direct quantitative comparison of the models on all validation datasets. MARE and MRE values for Model 1D, Raoult's law, UNIFAC, and REFPROP have been added to Sections 4.2 and 4.3. This significantly increases transparency and allows the reader to independently evaluate the performance of the models.
4. Validation on distillation data. The authors rightly noted that distillate composition is more indicative for validating the vapor pressure model than the distillation curve, which is strongly influenced by the process model (the number of theoretical plates). The text clearly demonstrates that Model 1D predicts distillate composition better than Raoult's law. This argument is compelling.
5. Error visualization. A shaded area (±5%) has been added to Figure 8, allowing for a visual assessment of the models' performance relative to a practical accuracy criterion. This is a small but useful improvement for data comprehension.
6. Deviation from ASTM D86. The authors added a discussion of how the lower distillate withdrawal rate in their experiments may have resulted in a greater degree of fractionation compared to the standard method. This is an important observation, increasing the reliability of the experimental data.
7. Physical interpretation of parameters. A meaningful discussion of the physical essence of the proposed modification of the Clausius-Clapeyron equation has been added. Reference to the packing of molecules in the condensed phase and the change in the effective activation energy of desorption in a heterogeneous environment gives the model a physical basis that goes beyond pure empirical evidence.